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The Method of High Accuracy Calculation of Robot Trajectory for the Complex Curves

The Method of High Accuracy Calculation of Robot Trajectory for the Complex Curves The geometric model accuracy is crucial for product design. More complex surfaces are represented by the ap- proximation methods. On the contrary, the approximation methods reduce the design quality. A new alternative calculation method is proposed. The new method can calculate both conical sections and more complex curves. The researcher is able to get an analytical solution and not a sequence of points with the destruction of the object semantics. The new method is based on permutation and other symmetries and should have an origin in the internal properties of the space. The classical method consists of finding transformation parameters for symmet- rical conic profiles, however a new procedure for parameters of linear transformations determination was ac- quired by another method. The main steps of the new method are theoretically presented in the paper. Since a double result is obtained in most stages, the new calculation method is easy to verify. Geometric modeling in the AutoCAD environment is shown briefly. The new calculation method can be used for most complex curves and linear transformations. Theoretical and practical researches are required additionally. Key words: analytical method, linear transformations, planar complex curves, symmetries INTRODUCTION and Bachmann. A hypothesis was put forward about space The mathematical calculations have always been the base built as a natural language text. of any engineering creations [1]. It takes a long time to de- To solve the characteristic equations, it is necessary to ac- velop exact mathematical methods and a human life is too cept the basis of these hypotheses. A classic solution is used short to get a solution. The first approximate calculation for their calculations. This solution follows from two postu- methods were suggested by the ancient Greeks. Newton lates. Then the type of the basic equation does not change proposed a canonical form for the detection of a complex and so it is possible to obtain the solution in an orthogonal curve on plane. He found canonical formulas for solving the way. The transformation matrix then has a great influence problem: on the change of the basic equation. However, everything changes the symmetry of space and therefore it is neces- + 2 +  +  + + = 0 (1) sary to use a non-orthogonal basis of the solution to solve + 3  + 3 + 3 + 6  + 3 + 3 + complicated processes of nonlinearity. (2) 3 +  = 0 The next stages of the study were related to the analysis of etc. But an analytical result was obtained single for the form symmetries in geometry. Symmetry were used for theore- (1) of the second stage in next times only by Euler. Thus, tical constructions by Euclid, but only mirror symmetry is Newton designed of the advanced approximation methods used in based researches us rule. Let us look at how Ne- [2] for form (2). wton's problem is solved now. Artificial intelligence methods have been used to solve this problem. The equivalence of the concepts of auto- REVIEW OF THE LITERATURE morphism in geometry and universal in linguistics was sho- The approximation theory is currently in the development wed by an analysis of the texts of the works of De Saussure stage. The authors [3, 4, 5, 6, 7] describe and develop the 248 Management Systems in Production Engineering 2020, Volume 28, Issue 4 theory of approximation, methods of approximation, algo- λ – scalars, is necessary for acquiring results in many areas rithms for data analysis, discrete approximation, from of science, such as physics, mechatronics, optimal manage- which the authors of this article draw the basic theoretical ment, cryptography, etc. background. Many authors are currently working on this is- The classical method consists of finding transformation pa- sue. Authors [8] solve the Apostol-Euler-Dunkl polynomials rameters for symmetrical conic profiles [29]. A new proce- with applications to series involving zeros of Bessel func- dure for parameters of linear transformations determina- tions, and author in [9] describe the growth of polynomials tion was acquired by another method [30, 31]. The Carte- outside of a compact set-The Bernstein-Walsh inequality sian product in the Euclidean plane R×R for reflections revisited, and further, the authors [10] deal with the R×R→R×R is the main object of the research. The specific approximation and Entropy Numbers of Embeddings Be- equation can be resolved by the application of new mathe- tween Approximation Spaces. matical methods by means of projective transformations in Many outputs of approximation techniques are widely used spaces with a large number of dimensions [32]. The resolu- in mechanics, economics, mechatronics, robotics [11, 12], tion is extraordinarily difficult for engineering calculations instrumentation, technology [13, 14, 15], assembly [16, 17], [33, 34, 35, 36, 37]. The goal of the authors was to find a informatics [18, 19] etc. The general disadvantage of all method that results in formulas without radical dependen- methods is that they have miscalculations. Obviously, the cies. Radical dependencies usually do not allow further an- exact computations accuracy bases on the ideal conditions alytical calculations. The new method should have an origin required for the project. in the internal properties of the space. Lower level headings From the point of view of the trajectory of the robot, we remain unnumbered; they are formatted as run-in head- recognize two basic tasks: the first is the task of the trajec- ings. tory of the mobile robot and the second is the trajectory of the robot effector. METHODOLOGY AND RESULTSOF RESEARCH Many authors deal with calculations, planning and simula- We focused our research approach on the creation of an tion of robot effector trajectories [20, 21, 22, 23, 24]. In this algorithm of complex curves linear transformation, as well case, the product precession depends on the tolerances of as on computer modeling with experimental research. the manufacturing. Even though the scientific scan for a systematic problem-solving in the theory was finished a Algorithmus of complex curves linear transformation century ago, some production conditions dictate more Let there be a trajectory by Ω – planar differentiable curve strict claims nowadays. The task to rise the industrial engi- in the Euclidean plane R×R. Curve is located in a Cartesian neering precision was set about fifty years ago. The main coordinate. Parametrical system defined by equations: objects of this research are the Jordan curves. The issue of  = (3) Jordan curves was dealt with by the authors in [25], where ! ! the main goal was to calculate the Jordan trajectory of ro- where: bot movement. The solutions to the characteristic equation x, y, t, k , k , ϵ R. x y for industrial robot's elliptic trajectories are designed in the  = A parametric system of equations  cannot be article [26].  = Other authors follow the trajectory of the mobile robot it- used. The equations define a line, thus, functions   ≠ self. Authors in the article [27] presents a novel method to and   ≠  at the same time. A system of parametric generate a trajectory for a mobile robot moving in an un- equations  or  can be composed certain environment where only a few way-points are ava-  = ! ! to solve any canonical equation from a Newton systematic. ilable to reach a nearby target state. The proposed method Let us do a linear transformation of (3) by the matrix  = interpolates way-points by the combination of a straight & & '' ' line and a fifth-order Bezier curve along which the curva- % (, & & ture varies continuously. where: In this paper [28] authors are proposed an algorithm that aij ϵ R. The parameters of the transformed figure must be calculates the suboptimal movement between two posi- obtained. tions, which automatically generates a cloud of safe via The calculation method for centrally symmetric conical sec- points around the workpiece and then by exploiting such tions was described in articles [26, 35, 36] earlier, therefore points finds the suboptimal safe path between the two po- we will not dwell on it. But for a complete understanding, sitions that minimizes movement time. we recommend that the reader refer to these materials, Mathematical linguistics and the theory of symmetries cre- since the angles α and β are taken from them. ate the theoretical base of this study. Authors of the men- The method is based on such a fundamental property of a tioned study obtained the first results at the beginning of plane (space) as symmetry. Symmetry has many definitions the 90's of the 20th century. They repeated the classical as it is used in various sciences. The Dieudonne definition is method of study in various ways. used in this study since it is the most modern postulate in The correct answer to the equation  ⃗ =  ⃗, analytic geometry. Dieudonne considered three types of where: 1 0 symmetries on the plane: the unitary matrix % (, the P – transformation matrix, 0 1 ⃗ – vector, −1 0 1 0 mirror matrixes % ( and % (, permutations 0 1 0 −1 A. LOZHKIN, P. BOZEK, K. MAIOROV – The Method of High Accuracy Calculation of Robot Trajectory… 249 0 1 −1 0 % (. The unitary matrix does not change the Euclidean 6. % ( will be transformed into a parametric trajec- 1 0 0 1 plane. Mirror symmetries determine the direction of the tory equations system in the form of  from (3) numerical axes. The first angle α is the corner of rotation of the quadratic and if the system has an alternative description into trans- 1 0 form by classic theory and the second angle β is the corner formation % (. This step of the transformation chain 0 −1 by permutation symmetry. A non-orthogonal basis can be needs to be investigated in more detail. Since it is necessary considered based on these angles. to take into account the characteristic equation separation. The basis was used to define a direct method for linear Additional research is needed. However, the algorithm transformations of central symmetrical conic sections by should be correct for many calculations. two angles. Further research with scale and transforma- 7. By rotating the curve by its own angle α, the analytical tions of rotation proved the method to be non-applicable formula of the transformed complex curve is found. for an exact transformation, but to coincide with classical The last research was building by parabola with equations results in the neighborhood. The transformations in que- 1 0 stion do not have a non-orthogonal basis. Own angles α1 ≠ system 8 by step 6. Matrix % ( must execute if 0 −1 α for calculations of compression along the axis, if the an- signature angles α and β are negative and matrix must exe- gle is not zero, results are similar as to results obtained by −1 0 cute % ( if signature angles α and β are positive. Ex- a classical method. A non-orthogonal basis coinciding with 0 1 the orthogonal basis does exist, no other non-orthogonal periments show that the trajectory obeys the laws of quan- bases are present. tum physics than classical analytical geometry additionally. The biggest distinction to be found was for the process of singular transformations, where by applying a classical met- Computer modeling and natural experiments hod each curve had to be transformed individually, the new The method of geometric modeling was used to verify the method in contrast offered eight groups of transformations found patterns. The experiments were carried out in the (independent of shape of the curve) [26]. AutoLisp language in AutoCAD 2007 environment. The re- + , −+ , + , quested transformation parameter: let us have curve by Four transformations % (, % (, % (, + −, + , −, + parametrical equations system (3). The curve is drawing by + , % ( were found for the deviation of the own angle, the system of equations: −, + = &    + & ''   ' ! ! where β ϵ {-π/2, 0, π/2}. In this case two non-orthogonal 9 (4) = &    + & '    ! ! bases can be found by a classical method, each coinciding with an orthogonal basis. Exact parameters are generated as a sequence of thickened straight-line segments with the new method. Digital experiments made it possible to black color. The parameters of the curve are obtaining the obtain a chain of transformations for finding the analytical by the new method Another curve is drawing based on the formula of the transformed curve. calculation results as a sequence of thin segments of 1. The transformation matrix is split into a product of two straight lines in red. A hypothesis is putting forward if the matrices: second curve is completely inside the first. & 0 The experiments were carried out for next curves from (4): '' = . / 0 & = 234 astroid by system  , 1 & /& = 46, and  = . /. & /& 1 ' '' where: 1 & /& '   = 2 234  1 + 234 2. The transformation . / . / pa- t ϵ [0, 2π]; cardiode by system  , & /& 1 ' '' ! = 2 46,  1 + 234 rameters (3) are discovered as a centrally symmetric conic where: section, where α is the angle of first ort (own angle); β is the  = t ϵ [0, 2π]; parabola by system 8 , angle of second ort (angle of permutation symmetry); where: and  are both scalars of the characteristic equation. = 234 = & '' t ϵ [-2, 2]; circle by system 8 , 3. New curve description system:  is con- = 46, = & ! ! where: sidered from (3). ;<= ;?' = & 4. An inverse transformation [37] T depending on β is per- ;<= ;@' t ϵ [0, 2π]; strofoid by system : , formed. If the value of β will be negative, the transfor- ;<= ;?' = & &, 234 5 − 46, 5 ;<= ;@' mation will result in 1 = % (, if the value where: 234 7 46, 7 − 234 5 46, 5 t ϵ [0, π]; hypotrohoid by system will be positive, 1 = % (, in this manner a 234 7 46, 7 = 2 234  + 5 234 transition from a non-orthogonal basis to an orthogonal ba- : , sis can be achieved. = 2 46,  − 5 234 5. The curve scalars b will be multiplied by coefficients where: and  . t ϵ [0, 6π]. 250 Management Systems in Production Engineering 2020, Volume 28, Issue 4 The formulas were taken from books [38, 39]. The results are shown in Fig. 1-6. Fig. 6 Parabola The characteristic equation  ⃗ =  ⃗′ is divided into four Fig. 1 Hypocycloid with four cusps types  ⃗′ C 8% ( , % ( , % ( , % (E. Division was not taken into account in previous studies. Obviously, this division has a significant impact on step 6 of the transformation chain. Therefore, this step is the most difficult at this stage of the researches. Kinematic pairs with simple paths are used to design most robot designs now. Newton proposed a complex classifica- tion of curves specifically for the design of robots with com- plex trajectories. Designing a robot with a complex trajec- tory is possible using approximate methods now, and desi- gning precise robots is not possible. Fig. 2 Cardioid RESULTS OF RESEARCH The results of theoretical and modeling research can be used in the field of robotics, mainly due to its accuracy of trajectory calculation and rendering, but also by reverse va- lidation of motion control of monitored robot effectors. The proposed model is applicable for simulation of a two- wheel self-balancing mobile robotic system or other unsta- ble robotic vehicles with parallel wheel alignment in nature. Robotic devices that belong to this category of robotic sys- tems re-quire some degree autonomous behavior, descri- bed in literature as a mobile service robot. A convenient ad- dition to the precise regulation and control of mobile se- Fig. 3 Strofoid rvice robots is the implementation of inertia navigation, which provides reverse validation and control, for example for a tilting system implemented on a two-wheel self-balan- cing mobile robotic system. Table 1 The experiments results Curve Matrix Angle β° Angle α° Scalar k k x y 0.759, 0.5 −1 Astroid (Fig. 1) % ( 15.13 75.87 2.270 1 0.7 1.587, 0.2 0.2 Cardioid (Fig. 2) % ( 47.38 42.61 Fig. 4 Circle 0.420 0.1 0.3 1.267, 0.3 −0.3 Strofoid (Fig. 3) % ( -62.66 -27.34 0.2 0.4 1.184 1.614, 1.5 −1 Circle (Fig. 4) % ( -52.02 -37.98 −0.5 1.2 0.447 Hypotrochoid 1.167, 0.3 −0.1 % ( -56.31 -33.69 (Fig. 5) 0.2 0.2 1.0837 1.000, 0.01 −0.6 Parabola (Fig. 6) % ( -1 -89 0.6 0.8 0.493 SUMMARY Engineering computations define the material life of the person. The precision of the calculation has an impact on Fig. 5 Hypotrochoid the comfort of the entity as well as on the way of living. The A. LOZHKIN, P. BOZEK, K. MAIOROV – The Method of High Accuracy Calculation of Robot Trajectory… 251 basic method for gaining exact engineering calculations is [12] M. Baranov, P. Bozek, V. Prajova, T. Ivanova, D. Novo- kshonov, A. Korshunov. “Constructing and calculating of the inverse matrix method. However, this method contains multistage sucker rod string according to reduced stress.:” one defect: it is applicable only to conical sections. Other Acta Montanistica Slovaca, vol. 22, pp. 107-115, Feb. 2017. sections are impossible to apply with the inverse matrix [13] T. Vopat, J. Peterka, V. Šimna. “The influence of different method. For this reason, Newton proposed a new curves types of copy milling on the surface roughness and tool life classification. Even though this proposal solved the defect, th of end mills.”25 DAAAM international symposium on intel- the calculation methods were not found. ligent manufacturing and automation, 2014. 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Alexander Lozhkin ORCID ID: 0000-0001-9048-2469 Kalashnikov Izhevsk State Technical University Institute of Informatics and Hardware Software department 30 let Pobedy, 5, Izhevsk, Russian Federation e-mail: lag.izh@gmail.com Pavol Bozek ORCID ID: 0000-0002-3891-3847 Slovak University of Technology in Bratislava Faculty of Materials Science and Technology Institute of Production Technologies Jana Bottu, 2781/25, 917 24 Trnava, Slovak Republic e-mail: pavol.bozek@stuba.sk Konstantin Maiorov ORCID ID: 0000-0003-4285-7697 Kalashnikov Izhevsk State Technical University Institute of Informatics and Hardware Software department 30 let Pobedy, 5, Izhevsk, Russian Federation e-mail: gibiskusus@gmail.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Management Systems in Production Engineering de Gruyter

The Method of High Accuracy Calculation of Robot Trajectory for the Complex Curves

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Abstract

The geometric model accuracy is crucial for product design. More complex surfaces are represented by the ap- proximation methods. On the contrary, the approximation methods reduce the design quality. A new alternative calculation method is proposed. The new method can calculate both conical sections and more complex curves. The researcher is able to get an analytical solution and not a sequence of points with the destruction of the object semantics. The new method is based on permutation and other symmetries and should have an origin in the internal properties of the space. The classical method consists of finding transformation parameters for symmet- rical conic profiles, however a new procedure for parameters of linear transformations determination was ac- quired by another method. The main steps of the new method are theoretically presented in the paper. Since a double result is obtained in most stages, the new calculation method is easy to verify. Geometric modeling in the AutoCAD environment is shown briefly. The new calculation method can be used for most complex curves and linear transformations. Theoretical and practical researches are required additionally. Key words: analytical method, linear transformations, planar complex curves, symmetries INTRODUCTION and Bachmann. A hypothesis was put forward about space The mathematical calculations have always been the base built as a natural language text. of any engineering creations [1]. It takes a long time to de- To solve the characteristic equations, it is necessary to ac- velop exact mathematical methods and a human life is too cept the basis of these hypotheses. A classic solution is used short to get a solution. The first approximate calculation for their calculations. This solution follows from two postu- methods were suggested by the ancient Greeks. Newton lates. Then the type of the basic equation does not change proposed a canonical form for the detection of a complex and so it is possible to obtain the solution in an orthogonal curve on plane. He found canonical formulas for solving the way. The transformation matrix then has a great influence problem: on the change of the basic equation. However, everything changes the symmetry of space and therefore it is neces- + 2 +  +  + + = 0 (1) sary to use a non-orthogonal basis of the solution to solve + 3  + 3 + 3 + 6  + 3 + 3 + complicated processes of nonlinearity. (2) 3 +  = 0 The next stages of the study were related to the analysis of etc. But an analytical result was obtained single for the form symmetries in geometry. Symmetry were used for theore- (1) of the second stage in next times only by Euler. Thus, tical constructions by Euclid, but only mirror symmetry is Newton designed of the advanced approximation methods used in based researches us rule. Let us look at how Ne- [2] for form (2). wton's problem is solved now. Artificial intelligence methods have been used to solve this problem. The equivalence of the concepts of auto- REVIEW OF THE LITERATURE morphism in geometry and universal in linguistics was sho- The approximation theory is currently in the development wed by an analysis of the texts of the works of De Saussure stage. The authors [3, 4, 5, 6, 7] describe and develop the 248 Management Systems in Production Engineering 2020, Volume 28, Issue 4 theory of approximation, methods of approximation, algo- λ – scalars, is necessary for acquiring results in many areas rithms for data analysis, discrete approximation, from of science, such as physics, mechatronics, optimal manage- which the authors of this article draw the basic theoretical ment, cryptography, etc. background. Many authors are currently working on this is- The classical method consists of finding transformation pa- sue. Authors [8] solve the Apostol-Euler-Dunkl polynomials rameters for symmetrical conic profiles [29]. A new proce- with applications to series involving zeros of Bessel func- dure for parameters of linear transformations determina- tions, and author in [9] describe the growth of polynomials tion was acquired by another method [30, 31]. The Carte- outside of a compact set-The Bernstein-Walsh inequality sian product in the Euclidean plane R×R for reflections revisited, and further, the authors [10] deal with the R×R→R×R is the main object of the research. The specific approximation and Entropy Numbers of Embeddings Be- equation can be resolved by the application of new mathe- tween Approximation Spaces. matical methods by means of projective transformations in Many outputs of approximation techniques are widely used spaces with a large number of dimensions [32]. The resolu- in mechanics, economics, mechatronics, robotics [11, 12], tion is extraordinarily difficult for engineering calculations instrumentation, technology [13, 14, 15], assembly [16, 17], [33, 34, 35, 36, 37]. The goal of the authors was to find a informatics [18, 19] etc. The general disadvantage of all method that results in formulas without radical dependen- methods is that they have miscalculations. Obviously, the cies. Radical dependencies usually do not allow further an- exact computations accuracy bases on the ideal conditions alytical calculations. The new method should have an origin required for the project. in the internal properties of the space. Lower level headings From the point of view of the trajectory of the robot, we remain unnumbered; they are formatted as run-in head- recognize two basic tasks: the first is the task of the trajec- ings. tory of the mobile robot and the second is the trajectory of the robot effector. METHODOLOGY AND RESULTSOF RESEARCH Many authors deal with calculations, planning and simula- We focused our research approach on the creation of an tion of robot effector trajectories [20, 21, 22, 23, 24]. In this algorithm of complex curves linear transformation, as well case, the product precession depends on the tolerances of as on computer modeling with experimental research. the manufacturing. Even though the scientific scan for a systematic problem-solving in the theory was finished a Algorithmus of complex curves linear transformation century ago, some production conditions dictate more Let there be a trajectory by Ω – planar differentiable curve strict claims nowadays. The task to rise the industrial engi- in the Euclidean plane R×R. Curve is located in a Cartesian neering precision was set about fifty years ago. The main coordinate. Parametrical system defined by equations: objects of this research are the Jordan curves. The issue of  = (3) Jordan curves was dealt with by the authors in [25], where ! ! the main goal was to calculate the Jordan trajectory of ro- where: bot movement. The solutions to the characteristic equation x, y, t, k , k , ϵ R. x y for industrial robot's elliptic trajectories are designed in the  = A parametric system of equations  cannot be article [26].  = Other authors follow the trajectory of the mobile robot it- used. The equations define a line, thus, functions   ≠ self. Authors in the article [27] presents a novel method to and   ≠  at the same time. A system of parametric generate a trajectory for a mobile robot moving in an un- equations  or  can be composed certain environment where only a few way-points are ava-  = ! ! to solve any canonical equation from a Newton systematic. ilable to reach a nearby target state. The proposed method Let us do a linear transformation of (3) by the matrix  = interpolates way-points by the combination of a straight & & '' ' line and a fifth-order Bezier curve along which the curva- % (, & & ture varies continuously. where: In this paper [28] authors are proposed an algorithm that aij ϵ R. The parameters of the transformed figure must be calculates the suboptimal movement between two posi- obtained. tions, which automatically generates a cloud of safe via The calculation method for centrally symmetric conical sec- points around the workpiece and then by exploiting such tions was described in articles [26, 35, 36] earlier, therefore points finds the suboptimal safe path between the two po- we will not dwell on it. But for a complete understanding, sitions that minimizes movement time. we recommend that the reader refer to these materials, Mathematical linguistics and the theory of symmetries cre- since the angles α and β are taken from them. ate the theoretical base of this study. Authors of the men- The method is based on such a fundamental property of a tioned study obtained the first results at the beginning of plane (space) as symmetry. Symmetry has many definitions the 90's of the 20th century. They repeated the classical as it is used in various sciences. The Dieudonne definition is method of study in various ways. used in this study since it is the most modern postulate in The correct answer to the equation  ⃗ =  ⃗, analytic geometry. Dieudonne considered three types of where: 1 0 symmetries on the plane: the unitary matrix % (, the P – transformation matrix, 0 1 ⃗ – vector, −1 0 1 0 mirror matrixes % ( and % (, permutations 0 1 0 −1 A. LOZHKIN, P. BOZEK, K. MAIOROV – The Method of High Accuracy Calculation of Robot Trajectory… 249 0 1 −1 0 % (. The unitary matrix does not change the Euclidean 6. % ( will be transformed into a parametric trajec- 1 0 0 1 plane. Mirror symmetries determine the direction of the tory equations system in the form of  from (3) numerical axes. The first angle α is the corner of rotation of the quadratic and if the system has an alternative description into trans- 1 0 form by classic theory and the second angle β is the corner formation % (. This step of the transformation chain 0 −1 by permutation symmetry. A non-orthogonal basis can be needs to be investigated in more detail. Since it is necessary considered based on these angles. to take into account the characteristic equation separation. The basis was used to define a direct method for linear Additional research is needed. However, the algorithm transformations of central symmetrical conic sections by should be correct for many calculations. two angles. Further research with scale and transforma- 7. By rotating the curve by its own angle α, the analytical tions of rotation proved the method to be non-applicable formula of the transformed complex curve is found. for an exact transformation, but to coincide with classical The last research was building by parabola with equations results in the neighborhood. The transformations in que- 1 0 stion do not have a non-orthogonal basis. Own angles α1 ≠ system 8 by step 6. Matrix % ( must execute if 0 −1 α for calculations of compression along the axis, if the an- signature angles α and β are negative and matrix must exe- gle is not zero, results are similar as to results obtained by −1 0 cute % ( if signature angles α and β are positive. Ex- a classical method. A non-orthogonal basis coinciding with 0 1 the orthogonal basis does exist, no other non-orthogonal periments show that the trajectory obeys the laws of quan- bases are present. tum physics than classical analytical geometry additionally. The biggest distinction to be found was for the process of singular transformations, where by applying a classical met- Computer modeling and natural experiments hod each curve had to be transformed individually, the new The method of geometric modeling was used to verify the method in contrast offered eight groups of transformations found patterns. The experiments were carried out in the (independent of shape of the curve) [26]. AutoLisp language in AutoCAD 2007 environment. The re- + , −+ , + , quested transformation parameter: let us have curve by Four transformations % (, % (, % (, + −, + , −, + parametrical equations system (3). The curve is drawing by + , % ( were found for the deviation of the own angle, the system of equations: −, + = &    + & ''   ' ! ! where β ϵ {-π/2, 0, π/2}. In this case two non-orthogonal 9 (4) = &    + & '    ! ! bases can be found by a classical method, each coinciding with an orthogonal basis. Exact parameters are generated as a sequence of thickened straight-line segments with the new method. Digital experiments made it possible to black color. The parameters of the curve are obtaining the obtain a chain of transformations for finding the analytical by the new method Another curve is drawing based on the formula of the transformed curve. calculation results as a sequence of thin segments of 1. The transformation matrix is split into a product of two straight lines in red. A hypothesis is putting forward if the matrices: second curve is completely inside the first. & 0 The experiments were carried out for next curves from (4): '' = . / 0 & = 234 astroid by system  , 1 & /& = 46, and  = . /. & /& 1 ' '' where: 1 & /& '   = 2 234  1 + 234 2. The transformation . / . / pa- t ϵ [0, 2π]; cardiode by system  , & /& 1 ' '' ! = 2 46,  1 + 234 rameters (3) are discovered as a centrally symmetric conic where: section, where α is the angle of first ort (own angle); β is the  = t ϵ [0, 2π]; parabola by system 8 , angle of second ort (angle of permutation symmetry); where: and  are both scalars of the characteristic equation. = 234 = & '' t ϵ [-2, 2]; circle by system 8 , 3. New curve description system:  is con- = 46, = & ! ! where: sidered from (3). ;<= ;?' = & 4. An inverse transformation [37] T depending on β is per- ;<= ;@' t ϵ [0, 2π]; strofoid by system : , formed. If the value of β will be negative, the transfor- ;<= ;?' = & &, 234 5 − 46, 5 ;<= ;@' mation will result in 1 = % (, if the value where: 234 7 46, 7 − 234 5 46, 5 t ϵ [0, π]; hypotrohoid by system will be positive, 1 = % (, in this manner a 234 7 46, 7 = 2 234  + 5 234 transition from a non-orthogonal basis to an orthogonal ba- : , sis can be achieved. = 2 46,  − 5 234 5. The curve scalars b will be multiplied by coefficients where: and  . t ϵ [0, 6π]. 250 Management Systems in Production Engineering 2020, Volume 28, Issue 4 The formulas were taken from books [38, 39]. The results are shown in Fig. 1-6. Fig. 6 Parabola The characteristic equation  ⃗ =  ⃗′ is divided into four Fig. 1 Hypocycloid with four cusps types  ⃗′ C 8% ( , % ( , % ( , % (E. Division was not taken into account in previous studies. Obviously, this division has a significant impact on step 6 of the transformation chain. Therefore, this step is the most difficult at this stage of the researches. Kinematic pairs with simple paths are used to design most robot designs now. Newton proposed a complex classifica- tion of curves specifically for the design of robots with com- plex trajectories. Designing a robot with a complex trajec- tory is possible using approximate methods now, and desi- gning precise robots is not possible. Fig. 2 Cardioid RESULTS OF RESEARCH The results of theoretical and modeling research can be used in the field of robotics, mainly due to its accuracy of trajectory calculation and rendering, but also by reverse va- lidation of motion control of monitored robot effectors. The proposed model is applicable for simulation of a two- wheel self-balancing mobile robotic system or other unsta- ble robotic vehicles with parallel wheel alignment in nature. Robotic devices that belong to this category of robotic sys- tems re-quire some degree autonomous behavior, descri- bed in literature as a mobile service robot. A convenient ad- dition to the precise regulation and control of mobile se- Fig. 3 Strofoid rvice robots is the implementation of inertia navigation, which provides reverse validation and control, for example for a tilting system implemented on a two-wheel self-balan- cing mobile robotic system. Table 1 The experiments results Curve Matrix Angle β° Angle α° Scalar k k x y 0.759, 0.5 −1 Astroid (Fig. 1) % ( 15.13 75.87 2.270 1 0.7 1.587, 0.2 0.2 Cardioid (Fig. 2) % ( 47.38 42.61 Fig. 4 Circle 0.420 0.1 0.3 1.267, 0.3 −0.3 Strofoid (Fig. 3) % ( -62.66 -27.34 0.2 0.4 1.184 1.614, 1.5 −1 Circle (Fig. 4) % ( -52.02 -37.98 −0.5 1.2 0.447 Hypotrochoid 1.167, 0.3 −0.1 % ( -56.31 -33.69 (Fig. 5) 0.2 0.2 1.0837 1.000, 0.01 −0.6 Parabola (Fig. 6) % ( -1 -89 0.6 0.8 0.493 SUMMARY Engineering computations define the material life of the person. The precision of the calculation has an impact on Fig. 5 Hypotrochoid the comfort of the entity as well as on the way of living. The A. LOZHKIN, P. 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Alexander Lozhkin ORCID ID: 0000-0001-9048-2469 Kalashnikov Izhevsk State Technical University Institute of Informatics and Hardware Software department 30 let Pobedy, 5, Izhevsk, Russian Federation e-mail: lag.izh@gmail.com Pavol Bozek ORCID ID: 0000-0002-3891-3847 Slovak University of Technology in Bratislava Faculty of Materials Science and Technology Institute of Production Technologies Jana Bottu, 2781/25, 917 24 Trnava, Slovak Republic e-mail: pavol.bozek@stuba.sk Konstantin Maiorov ORCID ID: 0000-0003-4285-7697 Kalashnikov Izhevsk State Technical University Institute of Informatics and Hardware Software department 30 let Pobedy, 5, Izhevsk, Russian Federation e-mail: gibiskusus@gmail.com

Journal

Management Systems in Production Engineeringde Gruyter

Published: Dec 1, 2020

Keywords: analytical method; linear transformations; planar complex curves; symmetries

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